Perpendicular axis theorem

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Description:

In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis at right angles to the plane, given the moments of inertia of the object about two perpendicularaxes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes <math>x,</math>, <math>y,</math>, and <math>z,</math> (which meet at origin <math>O,</math>) so that the body lies in the <math>xy,</math> plane, and the <math>z,</math> axis is perpendicular to the plane of the body. Let I<sub>x</sub>, I<sub>y</sub> and I<sub>z</sub> be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that

If a planar object (or prism, by the stretch rule) has rotational symmetry such that <math>I_x,</math> and <math>I_y,</math> are equal, then the perpendicular axes theorem provides the useful relationship:

<math>I_z = 2I_x = 2I_y,</math>

Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the <math>z,</math> axis is given...Read More